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given x³ + y³ + z³ = 3 xyz

(x+y+z)³

= (x+y)³ + z³ + 3 (x+y)z (x+y+z)

= (x³ + y³ + z³) + 3 x²y + 3 xy² + 3 x²z + 6 xyz + 3 xz² + 3 y²z + 3 yz²

replace the 6xyz term with 2 x³ + 2 y³ + 2 z³. Then rearrange the terms.

= 3 x³ + 3 y³ + 3 z³ + 3 x²y + 3 xy² + 3 x²z + 3 xz² + 3 y²z + 3 yz²

Bring terms with x² together. those with y² together.

= 3 x² (x+y+z) + 3 y² (y + x +z) + 3 z² (z + x +y)

= 3 (x + y + z) (x² + y² + z²)

x+y+z ≠ 0 as x,y,z>0 as log x, log y, log z exist.

hence, (x + y + z)² = 3 (x² + y² + z²) ---- (2)

simplify the above expression by expansion to get:

x² + y² + z² = xy + yz + xz

= (x+y+z)² / 3

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19) Here the logarithms are to the base 10.

Given number is expressed as N = x * 10^y like: 1.5015 * 10^5.

N = number. Log Characteristic Mantissa

N = x * 10^y Log N Floor[ Log N ] Log N - characteristic

y + Log x y Log x

Given, N = 0.00203 = 2.03 * 10^-3, and Log N = -3 + log 2.03

So characteristic is the exponent of 10: -3. Mantissa = Log 2.03

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20)

N = Log 3 = 0.4771 = 4.771 * 10^-1

Hence : characteristic is -1 and the mantissa is Log 4.771

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21)

Log x = 2 Log a + 3 Log b - 4 Log c

= Log a² + log b³ - log c⁴

= Log [ a² b³ / c⁴ ]

so x = a² b³ / c⁴

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22)

Here all the logarithms are to the base 1/2. Since all terms are for the same base, we dont need to write it.

Log x = Log 4 + Log 3 - Log 2

= Log (4 * 3 / 2)

= Log 6

x = 6